Well-Structured Problems for Multiplication and Division

The structure of multiplication problems

Grouping problems How many peanuts would the monkey eat if she ate 4 groups of peanuts with 3 peanuts in each group? The monkey ate 4 bags of peanuts. Each bag had the same number of peanuts in it. If the monkey ate 12 peanuts all together, how many peanuts were in each bag? (how many in each group?) The monkey ate some bags of peanuts. Each bag had 3 peanuts in it. Altogether the monkey ate 12 peanuts. How many bags of peanuts did the monkey eat? (how many groups?)

Rate problems A baby elephant gains 4 pounds each day. How many pounds will the baby elephant gain in 8 days? A baby elephant gains 4 pounds each day. How many days will it take the baby elephant to gain 32 pounds? A baby elephant gained 32 pounds in 8 days. If she gained the same amount of weight each day, how much did she gain in one day?

Price problems How much would 5 pieces of bubble gum cost if each piece costs 4 cents? If you bought 5 pieces of bubble gum for 20 cents, how much would each piece cost? If one piece of bubble gum costs 4 cents, how many can you buy for 20 cents?

Multiplicative comparison problems The first-grade class has a hamster and a gerbil. The hampster weighs 3 times as much as the gerbil. The gerbil weights 9 ounces. How much does the hamster weigh?

Array and Area problems (symmetric problems) Array problems help children understand that multiplication is commutative. For the second grade play, the chairs have been put into 4 rows with 6 chairs in each row. How many chairs have been put out for the play? (The monkey problem above is not solved with an array naturally, even though many books use arrays to show all kinds of grouping problems.) Area problems provide a basis for understanding multiplication and division of fractions. A baker has a pan of fudge that measures 8 inches on one side and 9 inches on another side. If the fudge is cut into square pieces 1 inch on each side, how many pieces of fudge does the pan hold?

Combination problems The Friendly Old Ice Cream Shop has 3 types of ice cream cones. They also have 4 flavors of ice cream. How many different combinations of an ice cream flavor and cone type can you get at the Friendly Old Ice Cream Shop?